How Does Bow Wave Angle Correlate with Boat Speed on a Catamaran?

[YellowPeril]

Hi

From this post you will deduce that one of my hobbies is sailing, I believe that applying a bit of science to the situation will improve my ability to make better tactical decisions on the spot.

I have a GPS but am interested in getting a quick visual of boat speed related to the angle that the bow wave makes to the boat. Is there a simple relation between bow wave angle and boat speed? The type of hull that I am interested is a catamaran hull (long and thin) so it can never plane and the speed range I am interested in is about from 10 to 20 knots (~20 km/h to 40 km/h)

Also I have heard in conversation that there is a maximum speed associated with a specific type of hull. (Hobie 16 catamaran for example) I would like to understand how this is determined, if it is at all possible. A hobie 16 hull does not have a centerboard like most hulls but is designed with an aerofoil shape. When one hull is flying (term used when one hull is raisd above the waterline), the other hull is in the water and the lift generates the stabalising influence (through hydrodynamic lift) that would normally be generated by a keel on a monohull. The hobie hull can be approximated by a long thin aerofoil dragged through the water in section. I think that there is a penalty incurred in the speed for this type of design, once again heard from somewhere. I would be interested to know what the lift and drag relations would be in relation to hull speed and how to determine them.

 

[fleem]

For a given wavelength, wave speed varies with water depth. So you'd have to know the water depth and you'd have to be able to detect at least one single frequency. More would add to your accuracy. This is certainly doable but it would probably take some fairly fancy image processing software. Once you know the spectrum (or at least one main frequency), angle, and water depth (assuming it doesn't vary much or your software knows the variations and accounts for them), then you could make a calculation for each wavelength speed using the equations in this wiki article:
http://en.wikipedia.org/wiki/Ocean_surface_wave#Science_of_waves

Then use trigonometry to figure boat speed (probably averaging multiple frequencies to add accuracy). I suppose in theory you might even see a hull signature in the spectrum and angles!

Its an interesting idea, but I think it would be hard to implement in practice.

 

[fleem]

Dang, I skimmed your post and thought you wanted to figure the speed of some distant boat! Ok -- watching your own wake makes it easier, and let's assume its nice deep water (much deeper compared to any wavelength you watch). In that case, pick a wavelength (preferably one much shorter than current water depth), calculate its speed from the simplified equation:

WaveSpeed = 1.25\sqrt{Wavelength}

(meters and seconds)

Then use this equation to calculate boat speed:

BoatSpeed = WaveSpeed * tan(\pi/2 - Angle)

Where Angle is the angle from the boat's path, in radians.

OK, that's the theory. However, I would seriously consider taking empirical data and recording it in a table of angle vs. speed! That way certain other unknowns are accounted for (like inaccuracies/distortions in your visual cortex, inaccuracy in recognizing a wavelength, shock waves, water makeup and temperature, etc.). Come to think of it, salt content would change things a lot, I think.

 

[YellowPeril]

Thanks Fleem,

Empirical measurements were what I had in mind. I just needed some background on the variables to see if it would be a worthwhile exercise. From what you have told me it could be worth the effort to make some crude measurements with the GPS while out on the water. Bearing in mind to repeat the exercise in different conditions, I should be able to get what I am after.

 

[Redbelly98]

https://www.physicsforums.com/latex_images/22/2294452-1.png

[/URL]

Alternatively,

V_boat = V_wave / tan(θ)​

is a somewhat more direct calculation (no need to subtract θ from π/2 or 90°)